Fractions

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 7 Fractions

Lesson objectives

By the end of this topic, you should be able to:

• Identify proper fractions, improper fractions and mixed numbers.
• Convert mixed numbers to improper fractions and back.
• Compare and order fractions.
• Perform the four basic operations on fractions and apply them to real situations.

Fractions

A fraction expresses part of a whole. The numerator counts the parts; the denominator says how many make the whole.

a) Three kinds

A proper fraction has a numerator smaller than its denominator, so its value is less than 1.

An improper fraction has a numerator equal to or larger than its denominator. Its value is 1 or more.

A mixed number is a whole number and a proper fraction written together.

b) Converting between them

To make a mixed number improper, multiply the whole part by the denominator. Add the numerator, and keep the same denominator.

To go back, divide the numerator by the denominator. The quotient is the whole part and the remainder is the new numerator.

So 13/5 gives 2 remainder 3, which is 2 3/5.

c) Equivalent fractions

Multiplying or dividing top and bottom by the same number does not change the value.

So 3/4, 6/8 and 9/12 are all the same amount.

This is what allows you to simplify, and to rename fractions to a common denominator.

d) Comparing and ordering

Give the fractions a common denominator, then compare the numerators.

Use the LCM of the denominators to keep the numbers small.

For 2/3, 3/4 and 5/8, the LCM of 3, 4 and 8 is 24. They become 16/24, 18/24 and 15/24. So the order is 5/8, 2/3, 3/4.

Give the answer using the original fractions.

e) Adding and subtracting

Find a common denominator, rename both fractions, then add or subtract the numerators only.

With mixed numbers, either convert to improper fractions first, or handle whole parts and fraction parts separately.

If a subtraction leaves the fraction part too small, borrow 1 from the whole part.

f) Multiplying

Multiply numerators together and denominators together. No common denominator is needed.

Convert any mixed numbers to improper fractions first.

Cancel common factors before multiplying. It keeps the numbers small and avoids a heavy simplification at the end.

g) Dividing

To divide by a fraction, multiply by its reciprocal.

Only the second fraction is inverted.

Convert mixed numbers to improper fractions before inverting. Inverting a mixed number as it stands gives nonsense.

h) Where this is used

Measuring in construction and tailoring. Sharing costs. Recipes. Any quantity that is not a whole number of units.

Words to know

• Numerator -- the top number, showing how many parts are taken.
• Denominator -- the bottom number, showing how many equal parts the whole has.
• Improper fraction -- one whose numerator is at least as large as its denominator.
• Reciprocal -- a fraction inverted, so the reciprocal of 4/9 is 9/4.
• Simplest form -- when numerator and denominator are coprime.

:::checkpoint Check yourself

1. Convert 3 2/7 to an improper fraction.
2. Convert 22/5 to a mixed number.
3. Work out 2/3 × 3/8.
4. Work out 1 1/2 ÷ 2/3. :::

Bridge to practice

The exercises begin with types of fraction and conversion, move through simplifying, comparing, and the four operations, and finish with word problems. Before any calculation, check whether the operation needs a common denominator — that single question determines the whole method.